Complexity and approximation results for bounded-size paths packing problems

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Complexity and approximation results for bounded-size paths packing problems

Jérôme Monnot, Sophie Toulouse

To cite this version:

Jérôme Monnot, Sophie Toulouse. Complexity and approximation results for bounded-size paths packing problems. Vangelis Th. Paschos. Combinatorial Optimization and Theoretical Computer Science Interfaces and Perspectives 30th anniversary of the LAMSADE, ISTE, not printed yet, 2007.

�hal-00152277�

Complexity and approximation results for bounded-size paths packing problems

1.1. Introduction

This chapter presents some recent works given by the authors ([MON 07a, MON 07b]) about the complexity and the approximation of several problems on computing col- lections of (vertex)-disjoint paths of bounded size.

1.1.1. Bounded-size paths packing problems

A P

k

partition of the vertex set of a simple graph G = (V, E) is a partition of V into q subsets V

1

, · · · , V

q

, each of size | V

i

| = k, such that the subgraph G[V

i

] indu- ced by any V

i

contains a Hamiltonian path. In other words, the partition (V

1

, . . . , V

q

) describes a collection of | V | /k vertex disjoint simple paths of length k − 1 (or, equiva- lently, simple paths on k vertices) on G. The decision problem called P

k

partitioning problem ( P

k

P

ARTITION

in short) consists, given a simple graph G = (V, E) on k × n vertices, in deciding whether G admits or not such a partition. The analogous problem where the subgraph G[V

i

] induced by V

i

is isomorphic to P

k

(the chordless path on k vertices) will be denoted by

INDUCED

P

k

P

ARTITION

. These two problems are NP-complete for any k ≥ 3, and polynomial otherwise, [GAR 79, KIR 78]. In fact, they both are a particular case of a more general problem called partition into isomorphic subgraphs, [GAR 79]. In [KIR 78], Kirkpatrick and Hell give a neces- sary and sufficient condition for the NP-completeness of the partition into isomorphic

Chapitre rédigé par Jérôme M

ONNOT

and Sophie T

OULOUSE

.

1

subgraphs problem in general graphs. P

k

P

ARTITION

has been widely studied in the literature, mainly because of its closeness to two famous optimization problems, na- mely : the minimum k-path partition problem (denoted by M

IN

k-P

ATH

P

ARTITION

) and the maximum P

k

packing problem (denoted by M

AX

P

k

P

ACKING

).

On the one hand, M

IN

k-P

ATH

P

ARTITION

can be viewed as an optimization ver- sion of P

k

P

ARTITION

where the constrainst on the exact length of the paths is relaxed.

M

IN

k-P

ATH

P

ARTITION

consists in partitioning the vertex set of a graph G = (V, E) into the smallest number of paths so that each path has at most k vertices (for ins- tance, M

IN

2-P

ATH

P

ARTITION

is equivalent to the maximum matching problem).

The optimal value is usually denoted by ρ

k−1

(G) for any k ≥ 2, by ρ(G) when no constraint occurs on the length of the paths (in particular, ρ(G) = 1 iff G has a Hamiltonian path). M

IN

k-P

ATH

P

ARTITION

has been extensively studied in the litera- ture, [STE 03, STE 00, YAN 97], and has applications in broadcasting problems (see for example [YAN 97]).

On the other hand, if we relax the exact covering constraint, then we obtain the op- timization problems M

AX

P

k

P

ACKING

and M

AX

I

NDUCED

P

k

P

ACKING

which consist, given a simple graph G = (V, E), in finding a maximum number of vertex-disjoint (induced) P

k

. When considering the weighted case (denoted by M

AX

W P

k

P

ACKING

and M

AX

WI

NDUCED

P

k

P

ACKING

, respectively), the input graph G = (V, E) is gi- ven together with a weight function w on its edges, and the goal is to find a col- lection P = { P

1

, . . . , P

q

} of vertex-disjoint (induced) P

k

that maximizes w( P ) = P

q

i=1

P

e∈Pi

w(e).

The special case of M

AX

W P

k

P

ACKING

where the graph is complete on k × n vertices is called the weighted P

k

partition problem ( P

k

P in short). In this case, each solution contains exactly n vertex disjoints paths of length k − 1. If the goal is to maxi- mize (M

AX

P

k

P), then we seek a P

k

partition of maximum weight, and if the goal is to minimize (M

IN

P

k

P), then we seek a P

k

partition of minimum weight. When consi- dering the minimization version, it is more often assumed that the instance is metric, i.e., that the weight function satisfies the triangle inequality : w(x, y) ≤ w(x, z) + w(z, y), ∀ x, y, z ; M

IN

M

ETRIC

P

k

P will refer to this restriction. Note that this latter version of the problem is closely related to the vehicle routing problem when restric- ting the route of each vehicle to at most k intermediate stops, [ARK 06, FRE 78]. Fi- nally, we also will consider the special case of metric instances where the weight func- tion is either 1 or 2 ; the corresponding problems will be denoted by M

AX

P

k

P

_1,2

and M

IN

P

k

P

1,2

( P

k

P

1,2

when the goal is not specified). Such a restriction makes sense, since it provides an alternative relaxation of the initial decision problem P

k

Partition ; moreover, M

IN

P

k

P

1,2

and M

IN

k-P

ATH

P

ARTITION

are strongly connected.

All theses problems are very closed one to each other. In particular, P

k

P

ARTITION

NP-completeness implies the NP-hardness of both M

IN

k-P

ATH

P

ARTITION

and P

k

P

(even when restricting to P

k

P

_1,2

) ; conversely, P

k

P

ARTITION

is polynomial-time de- cidable on instance families where M

IN

k-P

ATH

P

ARTITION

or M

AX

P

k

P

ACKING

are polynomial-time computable.

1.1.2. Complexity and approximability status

The minimum k-path partition problem is obviously NP-complete in general graphs [GAR 79], and remains intractable in comparability graphs, [STE 03], in cographs, [STE 00], and in bipartite chordal graphs, [STE 03] (when k is part of the input). Note that most of the proofs of NP-completeness actually establish the NP-completeness of P

k

P

ARTITION

. Nevertheless, the problem turns out to be polynomial-time solvable in trees, [YAN 97], in cographs when k is fixed, [STE 00] and in bipartite permutation graphs, [STE 03]. Note that one can also find in the literature several results about the problem that consists in partitioning the graph into disjoints paths of length at least 2, [WAN 94, KAN 03].

This chapter proposes new complexity and inapproximability results for (

INDU

-

CED

) P

k

P

ARTITION

, M

IN

k-P

ATH

P

ARTITION

and M

AX

(W)(I

NDUCED

) P

k

P

ACKING

, mostly in the case of bipartite graphs, discussing the graph maximum degree. Namely, we study the case of bipartite graphs of maximum degree 3 : first, these problems are NP-complete for any k ≥ 3 (and this even if the graph is planar, for k = 3) ; second, there is no PTAS for M

AX

(

INDUCED

) P

k

P

ACKING

or, more precisely, there is a constant ε

k

> 0 such that it is NP-hard to decide whether a maximum (induced) P

k

-packing is of size n or of size upper bounded by (1 − ε

k

)n. On the opposite side, all these problems trivially become polynomial-time computable both in graphs of maximum degree 2 and in forests.

Where these problems are intractable, what about their approximation level ? We recall that a given problem is said to be ε-approximable if it admits an algorithm that polynomially computes on any instance a solution that is at least (if maximizing, at most if minimizing) ε times the optimum value. To our knowledge, there is no specific approximation result for neither M

IN

k-P

ATH

P

ARTITION

, nor M

AX

W P

k

P

ACKING

, in general graphs. Nevertheless, one can find some approximation results for the k- path partition problem where the objective consists in maximizing the number of edges of the paths that participate to the solution (see [VIS 92] for the general case, [CSA 02] for dense graphs). Concerning M

AX

W P

k

P

ACKING

, using approximation results for the maximum weighted k-packing problem (mainly based on local search techniques), [ARK 98], one can obtain a (

_k−1¹

− ε)-approximation ; in particular, M

AX

W P

3

P

ACKING

is (

¹₂

− ε)-approximable.

In the case of complete graphs, M

AX

P

k

P is standard-approximable for any k, [HAS 97]. In particular, M

AX

P

3

P and M

AX

P

4

P are respectively 35/67 − ε, [HAS 06]

and 3/4, [HAS 97] approximable. Note that for k = 2, a P

2

-partition is a perfect

matching and hence, M

IN

P

₂

P and M

AX

P

₂

P both are polynomial-time computable.

The minimum case is trickier : from the fact that P

k

P

ARTITION

is NP-complete in general graphs, it is NP-hard to approximate M

IN

P

k

P within 2

^p(n)

for any polyno- mial p, for any k ≥ 3. Nevertheless, one could expect that the metric instances are constant-approximable, even though no approximation rate (to our knowledge) has been established so far for M

IN

M

ETRIC

P

k

P.

Here, we provide new approximation results for M

IN

3-P

ATH

P

ARTITION

, M

AX

W- P

3

P

ACKING

and P

k

P. Concerning the two former problems, we propose a 3/2- approximation for M

IN

3-P

ATH

P

ARTITION

in general graphs and a 1/3 (resp., a 1/2)- approximation for M

AX

W P

3

P

ACKING

in general (resp., bipartite) graphs of maxi- mum degree 3. But we more focus on P

k

P, and more specifically on P

4

P, by ana- lyzing the performance of a specific algorithm proposed by Hassin and Rubinstein, [HAS 97], under different assumptions on the input. Doing so, we put to the fore the effectiveness of this algorithm by proving that it provides new approximation ratios for both standard and differential measures, for both maximization and minimization versions of the problem. But, before going so far, we briefly recall the basis of ap- proximation theory, introduce some notations and then give this outline of the chapter.

1.1.3. Theoritical framework, notations and organization

Consider an instance I of an NP-hard optimization problem Π and a polynomial- time algorithm A that computes feasible solutions for Π. Denote by apx

_Π

(I) the value of a solution computed by A on I, by opt

_Π

(I) the value of an optimal solution and by wor

Π

(I) the value of a worst solution (that corresponds to the optimum value when reversing the optimization goal). The quality of A is expressed by means of approximation ratios that somehow compare the approximate value to the optimum one. So far, two measures stand out from the literature : the standard ratio [AUS 99]

(the most widely used) and the differential ratio [AUS 80, BEL 95, DEM 96, HAS 01].

The standard ratio is defined by ρ

Π

(I, A ) = apx

_Π

(I)/opt

_Π

(I) if Π is a maximization problem, by ρ

Π

(I, A ) = opt

_Π

(I)/apx

_Π

(I) otherwise, whereas the differential ratio is defined by δ

Π

(I, A )= (wor

Π

(I) − apx

_Π

(I))/(wor

Π

(I) − opt

_Π

(I)). In other words, the standard ratio divides the approximate value by the optimum one, whereas the differential ratio divides the distance from a worst solution to the approximate value by the instance diameter.

Within the worst case analysis framework and given a universal constant ε ≤ 1 (resp., ε ≥ 1), an algorithm A is said to be an ε-standard approximation for a maxi- mization (resp. a minimization) problem Π if ρ

I,AΠ

(I) ≥ ε ∀ I (resp., ρ

_AΠ

(I) ≤ ε

∀ I). With respect to differential approximation, A is said to be ε-differential approxi-

mate for Π if δ

_AΠ

(I) ≥ ε, ∀ I, for a universal constant ε ≤ 1. Equivalently, seing

any solution value as a convex combination of the two values wor

Π

(I) and opt

_Π

(I),

an approximate solution value apx

_Π

(I) will be ε-differential approximate if for any

instance I, apx

_Π

(I) ≥ ε × opt

_Π

(I) + (1 − ε) × wor

Π

(I) (for the maximization case ; reverse the sense of the inequality when minimizing). For both measures, a given problem Π is said to be constant approximable if there exists a polynomial-time algo- rithm A and a universal constant ε such that A is an ε- approximation for Π. The class of problems that are standard- (resp., differential-) constant-approximable is denoted by APX (resp., by DAPX). If Π admits a polynomial-time approximation scheme, that is, a whole algorithm family (A

ε

)

_(ε)

such that A

ε

is ε-approximate for any ε (note that the time-complexity of A

ε

may be exponential in 1/ | 1 − ε | ), then Π belongs to the class PTAS (resp., DPTAS).

The notations that will be used are the usual ones according to graph theory. Mo- reover, we exclusively work in undirected simple graphs. In this chapter, we often identify a path P of length k − 1 with P

k

, even if P contains a chord. However, when dealing with

INDUCED

P

k

P

ARTITION

, the paths that are considered are chordless. Fi- nally, when no ambiguity occurs on the problem that is concerned, we will omit the reference to Π to denote the values apx(I), opt(I) and wor(I). For a better unders- tanding of what follows, we recall some basic concepts of graph theory : a simple graph G = (V, E) is said to be bipartite (or, equivalently, 2-colorable) if there exists a partition L, R of its vertex set such that E is contained in L × R. A graph is pla- nar if it can be drawn in the plane so that no edges intersect. A path (resp., a cycle) Γ = { v

j1

, . . . , v

jq

} ⊆ E in G of length at least 2 (resp., of length at least 4) is chord- less if there is in E no other edge than the ones of Γ linking two vertices of Γ. G is chordal if none of its cycle of length at least 4 is chordless. G is an interval graph if one can associate to each vertex v

j

∈ V an interval [a

j

, b

j

] on the real line such that two intervals [a

j

, b

j

] and [a

ℓ

, b

ℓ

] intersect iif the edge [v

j

, v

ℓ

] belongs to E ; note that interval graphs are special cases of chordal graphs.

This chapter is organized as follows : the two next sections are dedicated to the study of (

INDUCED

) P

k

P

ARTITION

, M

AX

(I

NDUCED

) P

k

P

ACKING

and M

IN

k-P

ATH

- P

ARTITION

. Section 1.2 focus on the complexity status of those problems in bipartite graphs, whereas Section 1.3 proposes some approximation results for M

AX

W P

3

P

AC

-

KING

and M

IN

3-P

ATH

P

ARTITION

. The fourth section is then dedicated to both stan- dard and differential approximation of P

k

P. Subsection 1.4.1 provides a differen- tial approximation for P

k

P while bridging some gap between differential approxima- tion of TSP and differential approximation of P

k

P. Finally, Subsection 1.4.2, which constitutes the main part of Section 1.4, leads a complete analysis of the approxima- tion level of an algorithm proposed by Hassin and Rubinstein [HAS 97], depending on the approximation measure that is considered and the characteristics of the input weight function.

The two main points of the chapter are, on the one hand, the establishment of new

complexity results concerning P

k

P

ARTITION

and related problems in bipartite graphs

by means of reductions (section 1.2) and, on the other hand, the way the algorithm that

is addressed in section 1.4.2 appears to be robust, in the sense that this latter provides

a

^i,1₃

a

^i,1₂

a

^i,2₃

a

^i,2₂

a

^i,3₃

a

^i,3₂

a

^i,11

a

^i,21

a

^i,31

Figure 1.1. The gadget H(c

i

) when c

i

is a 3-tuple.

good quality solutions (the best known so far), whatever version of the problem we deal with, whatever approximation framework within which we estimate the approxi- mate solutions.

1.2. Complexity of P

k

P

ARTITION

and related problems in bipartite graphs 1.2.1. Negative results from the k-dimensional matching problem

1.2.1.1. k-dimensional matching problem

The negative results we present all are based on a transformation from the k- dimensional matching problem, kDM, which is known to be NP-complete, [GAR 79].

An instance of kDM consists of a subset C = { c

1

, . . . , c

m

} ⊆ X

1

× . . . × X

k

of k-tuples, where X

1

, . . . , X

k

are k pairwise disjoint sets of size n. A matching is a sub- set M ⊆ C such that no two elements in M agree in any coordinate, and the purpose of kDM is to answer the question : does there exist a perfect matching M on C , that is, a matching of size n ? In its optimization version, the maximum k-dimensional matching problem (M

AX

kDM) addresses the question of computing a matching that is of maximal size.

1.2.1.2. Transforming an instance of kDM into an instance of P

k

P

ACKING

Let I = (X

1

, . . . , X

k

; C ) be an instance of kDM, where | X

q

| = n, ∀ q and |C| = m. We denote by X the union of the element sets X

1

, . . . , X

k

. Furthermore, for each element e

j

∈ X, we denote by d

^j

its degree, where the degree of an element e

j

is defined as the number of k-tuples c

i

∈ C that contain e

j

. We build an instance G = (V, E) of

INDUCED

P

k

P

ACKING

, where G is a bipartite graph of maximum degree 3, by associating a k-tuple gadget H(c

i

) to each k-tuple c

i

∈ C , an element gadget H(e

j

) to each element e

j

∈ X , and then by linking the two gadget families by some edges. Our construction (more precisely, the element gadgets) depends on the parity of k.

1) The element gadget H(c

i

). For any k-tuple c

i

∈ C , the gadget H (c

i

) consists of a collection

P

^i,1

, . . . , P

^i,k

of k vertex-disjoint P

k

with P

^i,q

= n

a

^i,q₁

, . . . , a

^i,q_k

o

for

q = 1, . . . , k, plus the edges [a

^i,q₁

, a

^i,q+1₁

] for q = 1 to k − 1. Hence, H (c

i

) contains

l

^j1

= v

^j1

v

_N^jj+1

l

^j₂

= v

₇^j

Figure 1.2. The gadget H (e

j

) for k = 3 and d

^j

= 2.

l

^j₁

= v

^j₁

v

^j_Nj+1

v

^j_Nj

l

^j2

= v

9^j

Figure 1.3. The gadget H (e

j

) for k = 4 and d

^j

= 2.

the k initial paths P

^i,1

, . . . , P

^i,k

, plus the additional path n

a

^i,1₁

, . . . , a

^i,k₁

o

. Figure 1.1 proposes an illustration of the k-tuple gadget when k = 3.

2) The element gadget H (e

j

). Let e

j

∈ X be an element, with degree d

^j

. We distin- guish two cases according to the parity of k.

– Odd values of k. H (e

j

) is defined as a cycle n

v

^j₁

, . . . , v

_N^jj+1

, v

₁^j

o

on N

^j

+ 1 vertices, where N

^j

= k(2d

^j

− 1). Moreover, for p = 1 to d

^j

, we denote by l

^j_p

the vertex of index 2k(p − 1) + 1. Thus, the element gadget is a cycle on a number of vertices that is a multiple of k plus 1, with d

^j

remarkable vertices l

^j_p

that will be linked to the k-tuple gadgets.

– Even values of k. In this case, N

^j

is also even and thus, a cycle on N

^j

+1 vertices may not be part of a bipartite graph. In order to fix that problem, we define H(e

j

) as a cycle n

v

₁^j

, . . . , v

_N^jj

, v

₁^j

o

on N

^j

vertices, plus an additional edge [v

^j_Nj

, v

_N^jj+1

]. The special vertices l

^j_p

still are defined as l

^j_p

= v

_2k(p−1)+1^j

for p = 1, · · · , d

^j

(note that l

^j_dj

never matches v

^j_Nj

). Figures 1.2 and 1.3 illustrate H (e

j

) for the couple of values k = 3, d

^j

= 2 and k = 4, d

^j

= 2, respectively.

3) Linking element gadgets to k-tuple gadgets. For any couple (e

j

, c

i

) such that e

j

is

the value of c

i

on the q-th coordinate, the two gadgets H (c

i

) and H (e

j

) are connected

using one of the edges [a

^i,q₂

, l

^j_pi

], p

i

∈ { 1, . . . , d

^j

} . The vertices l

_p^j_i

that will be linked

to a given gadget H (c

i

) must be chosen so that each vertex l

^j_p

from any gadget H (e

j

)

will be connected to exactly one gadget H (c

i

).

The described construction obviously leads to a graph G = (V, E) that is bipartite, of maximum degree 3, and such that every of the P

k

it contains is chordless. Its number of vertices is | V | = 3k

²

m + (1 − k)kn : consider, on the one hand, that each gadget H(c

i

) is a graph on k

²

vertices and, on the other hand, that P

kn

j=1

d

^j

= km (wlog., we may assume that each element e

j

appears at least once in C ).

1.2.1.3. Analyzing the obtained instance of P

k

P

ACKING

Let us define on G some remarkable P

k

packings on the vertex subsets V (H (c

i

)) and V (H(e

j

)).

P

k

packings on V (H (c

i

)), for i = 1, . . . , m :







P

ⁱ

= ∪

^kq=1

P

^i,q

∪ n

a

^i,1₁

, a

^i,2₁

, . . . , a

^i,k₁

o

with P

^i,q

= n

a

^i,q_k

, . . . , a

^i,q₂

, l

i,q

o ∀ q Q

ⁱ

= ∪

^kq=1

Q

^i,q

with Q

^i,q

= n

a

^i,q_k

, . . . , a

^i,q₂

, a

^i,q₁

o

∀ q (where l

i,q

denotes the vertex from some H(e

j

) that is linked to a

^i,q₂

)

P

k

packings on V (H (e

j

)), for j = 1, . . . , kn :

∀ p = 1, . . . , d

^j

, P

p^j

is defined as the only possible P

k

partition of V (H(e

j

)) \{ l

^j_p

}

Note that these collections are of size |P

ⁱ

| = k + 1 ∀ i, |Q

ⁱ

| = k ∀ i and |P

p^j

| = 2d

^j

− 1 ∀ j ∀ p ∈ { 1, . . . , d

^j

} . With the help of these packings, we now put to the fore three properties that will be the key of our further argumentation.

P

ROPERTY

1.–

(i) For any i, P

ⁱ

and Q

ⁱ

are the only two possible P

k

partitions of V (H(c

i

)).

(ii) Within a P

k

partition of V , and for any j = 1, . . . , kn, the collections P

1^j

, . . . , P

d^jj

are the only possible P

k

partitions of V (H (e

j

)).

(iii) Let P

^∗

be a maximum P

k

packing on G ; we can always assume the following : (iii.a) for any i, P

^∗

contains either the packing P

ⁱ

, or the packing Q

ⁱ

;

(iii.b) for any j, P

^∗

contains one of the packings P

p^j

, for some p.

P

ROOF

.– For sake of simplicity, we assume that k is odd, even though the arguments also hold for even values of k.

For (i). Quite immediate, from the observation that a given vertex a

^i,q_k

may only be covered by either P

^i,q

or Q

^i,q

.

For (ii). Let P be a P

k

partition of V and consider an element e

j

; since H (e

j

) contains N

^j

= k(2d

^j

− 1) + 1 vertices, at least one edge e of some P

ℓ

in P links H (e

j

) to a given H(c

i

), using an l

^j_p

vertex ; we deduce from the previous point that P

ℓ

is some P

^i,q

path and thus, that l

_p^j

is the only vertex of P

ℓ

that intersects H (e

j

).

Consider now any two vertices l

^j_p

and l

^j_p′

, p < p

^′

, from H (e

j

) ; the 2k(p

^′

− p) − 1

vertices that separate l

^j_p

and l

_p^j′

might not be covered by any collection of P

k

. Hence, exactly one l

^j_p

vertex of H(e

j

) is covered by some P

^i,q

and thus, P contains the corresponding P

k

packing P

p^j

.

For (iii.a). Any maximal size P

k

packing must use (at least) one of the two vertices a

^i,q₁

and l

i,q

, for any couple (i, q), where l

i,q

denotes the vertex from some H (e

j

) that is linked to a

^i,q₂

. Suppose the reverse, for some (i, q) : then, none of the vertices l

i,q

, a

^i,q₁

, a

^i,q₂

, . . . , a

^i,q_k

may be part of a path from P

^∗

and thus, P

^i,q

or Q

^i,q

could be added to P

^∗

, that would contradict the optimality of P

^∗

. If the edge [a

^i,q₁

, a

^i,q₂

] (resp., [a

^i,q₂

, l

i,q

] and not [a

^i,q₁

, a

^i,q₂

]) is used by some path P ∈ P

^∗

, then P can be replaced in P

^∗

by the path Q

^i,q

(resp., by P

^i,q

). If none of the edges [a

^i,q₁

, a

^i,q₂

] and [a

^i,q₂

, l

i,q

] are used by P

^∗

, replace by P

^i,q

(resp., by Q

^i,q

) the path from P

^∗

that uses l

i,q

(resp., a

^i,q₁

and not l

i,q

). At that point, the collection P

^∗

contains for any k-tuple c

i

at least k paths P

^i,q

and Q

^i,q

(one for each coordinate q = 1, . . . , k). Now, each time P

^∗

does not contain the packing P

ⁱ

, we replace these paths by the whole collection Q

ⁱ

. For (iii.b). Assume the reverse, for some element e

j

; that means that at least 2 vertices l

^j_pi

and l

^j_pi′

of H (e

j

) are used in P

^∗

by paths P

^i,q

and P

^i′,q′

, with p

i

< p

i^′

(or P

^∗

would not be of maximal size). Choose two consecutive such vertices, in the sense that P

^∗

does not use any of the paths P

^{i′′,q′′}

for l

^j_pi′′

such that p

i

< p

i^′′

< p

i^′

. Since there are 2k(p

i^′

− p

i

) − 1 vertices of H(e

j

) between l

_p^j_i

and l

^j_pi′

, we can replace P

^i,q

, P

^i′,q′

and the paths of P

^∗

between vertices l

^j_pi

and l

_p^j_i′

by P

^i,q

and 2(p

i^′

− p

i

) paths using vertices between l

_p^j_i

and l

_p^j

i′

, plus l

_p^j

i′

. Observe that, in such a case, the packing P

^i′

will be replaced in P

^∗

by the packing Q

^i′

, according to the previous property.

By repeating this procedure, we obtain a maximal size P

k

packing that fulfills the requirements of items (iii.a) and (iii.b).

1.2.1.4. NP-completeness and APX-hardness

T

HEOREM

1.– P

k

P

ARTITION

and

INDUCED

P

k

P

ARTITION

are NP-complete in bi- partite graphs of maximum degree 3, for any k ≥ 3.

As a consequence, M

AX

(

INDUCED

) P

k

P

ACKING

and M

IN

k-P

ATH

P

ARTITION

are NP- hard in bipartite graphs with maximum degree 3, for any k ≥ 3.

P

ROOF

.– Let I = (X

1

, . . . , X

k

; C ) and G = (V, E) be an instance of kDM and the graph produced by construction described is Subsection 1.2.1.2, respectively. First, we recall that any path of length k − 1 in G is chordless ; thus, the result holds for both P

k

P

ARTITION

and

INDUCED

P

k

P

ARTITION

. We claim that there exists a perfect matching M ⊆ C on I iff there exists a partition P of G into P

k

.

Let P be such a partition on G ; from Property 1 item (i), we know that each gad-

get H (c

i

) is covered by either P

ⁱ

or Q

ⁱ

. Moreover, Property 1 item (ii) indicates

that every gadget H (e

j

) is covered by some P

p^j

collection ; those two facts ensure

a^i,1¹ a^i,1² a^i,1³ a^i,1¹ a^i,1² a^i,1³

a^i,3¹ a^i,2¹ a^i,3² a^i,2² a^i,3³ a^i,2³ a^i,3¹ a^i,2¹ a^i,3² a^i,2² a^i,3³ a^i,2³

li,1 li,2 li,3 li,1 li,2 li,3

c

i

∈ M c

i

∈ / M

Figure 1.4. A vertex partition of a H (c

i

) gadget into 2-edge paths.

that exactly one H (c

i

) gadget for some k-tuple that contains e

j

is covered by a P

ⁱ

collection and therefore, the set M =

c

i

| P

ⁱ

⊆ P defines a perfect matching on I.

Conversely, let M be a perfect matching on C ; we build a packing P applying the following rule : if a given element c

i

belongs to M , then use P

ⁱ

to cover H (c

i

) ; use Q

ⁱ

otherwise (Figure 1.4 illustrates this construction for 3DM). Since M is a perfect matching, exactly one vertex l

^j_p

per gadget H(e

j

) is covered by some P

^i,q

. Thus, on a given cycle H(e

j

), the N

^j

= k(2d

^j

− 1) vertices that remain uncovered can be covered using the corresponding collection P

p^j

.

Thus, the construction is a Karp reduction, and from the NP-completeness of kDM, [GAR 79], we deduce the NP-completeness of (

INDUCED

) P

k

P

ARTITION

in bipartite graphs of maximum degree 3. However, by a more accurate observation, we actually may obtain a stronger result, for k = 3 ; namely, (

INDUCED

) P

3

P

ARTITION

NP-completeness still holds when restricting ourselves to planar instances. Indeed, on the one hand, the restriction P

LANAR

3DM of 3-dimensional matching to planar ins- tances still is NP-complete, [DYE 86] ; on the other hand, if the initial instance I of kDM is planar, then the graph G also is planar for an appropriate choice of the linking edges [a

^i,q₂

, l

i,q

].

T

HEOREM

2.– P

3

P

ARTITION

and

INDUCED

P

3

P

ARTITION

are NP-complete in pla- nar bipartite graphs with maximum degree 3.

As a consequence, M

AX

(

INDUCED

) P

3

P

ACKING

and M

IN

3-P

ATH

P

ARTITION

are NP- hard in planar bipartite graphs with maximum degree 3.

If we now turn to the optimization problems, we can observe that the construction described in Subsection 1.2.1.2 also enables to establish an APX-hardness result for the maximization problems M

AX

P

k

P

ACKING

and M

AX

(

INDUCED

) P

k

P

ACKING

. We consider the optimization version of kDM, denoted by M

AX

kDM, and the follo- wing inapproximability result : for any k ≥ 3, there is a constant ε

^′_k

> 0 such that

∀ I = (X

1

, . . . , X

k

; C ) instance of kDM with | X

1

| = · · · = | X

k

| = n, it is NP-hard

to decide between opt(I) = n and opt(I) ≤ (1 − ε

^′_k

)n, where opt(I) is the value of a maximum matching on C . This result also holds if we restrict ourselves to instances with bounded degree, namely, to instances I satisfying : ∀ j = 1, . . . , kn, d

^j

≤ f (k), where f (k) is a constant ; we refer to [PET 94] for k = 3 (where the result is proved with f (3) = 3), to [KAR 06] for other values of k.

T

HEOREM

3.– For any k ≥ 3, there is a constant ε

k

> 0, such that ∀ G = (V, E) instance of M

AX

(I

NDUCED

) P

k

P

ACKING

where G is a bipartite graph of maximum degree 3, it is NP-hard to decide between opt(G) =

^|V_k^|

and opt(G) ≤ (1 − ε

k

)

^|V_k^|

, where opt(G) is the value of a maximum (induced) P

k

-Packing on G.

P

ROOF

.– Let I = (X

1

, . . . , X

k

; C ) be an instance of kDM, with | X

q

| = n ∀ q and

|C| = m, such that the degree d

^j

of any element e

j

is bounded above by f (k).

Consider the graph G = (V, E) produced by the construction described in Subsec- tion 1.2.1.2 ; we recall that | V | = 3k

²

m − k

²

n + kn. Let (M

^∗

, P

^∗

) be a couple of optimal solutions on I and G, with values opt(I) and opt(G), respectively. From Property 1 items (iii.a) and (iii.b), we can assume that P

^∗

satisfies the following :

– for any i, P

^∗

contains either the packing P

ⁱ

, or the packing Q

ⁱ

; – for any j, P

^∗

contains one of the packings P

1^j

, . . . , P

d^jj

.

Hence, the set M = { c

i

∈ C : P

ⁱ

∈ P

^∗

} of k-tuples c

i

such that P

^∗

contains P

ⁱ

defines a matching on I ; moreover, the value opt(G) of P

^∗

can be expressed as :

opt(G) = (km + | M | ) +

kn

X

j=1

2d

^j

− 1

= 3km − kn + | M |

From | M | ≤ | M

^∗

| , we then deduce : opt(G) ≤ opt(I) + 3km − kn.

If opt(I) = n : we know from Theorem 1 that I has a perfect matching iff G admits a P

k

Partition, that is, opt(I) = n iif opt(G) =

^|V_k^|

= 3km − kn + n. Suppose now that opt(I) ≤ (1 − ε

^′_k

)n. Then, necessarily : opt(G) ≤ 3km − kn + (1 − ε

^′_k

)n = (3km − kn + n) − ε

^′_k

n. By setting ε

k

=

_3km−kn+nⁿ

ε

^′_k

, we obtain opt(G) ≤ (1 − ε

k

)(3km − kn +n). Finally, since d

^j

≤ f (k), we deduce that km ≤ kf(k)n and then, that ε

k

≥

3f(k)k−k+1¹

ε

^′_k

= O (1). In conclusion, deciding between opt(G) = | V | /n and opt(G) ≤ (1 − ε

k

) | V | /n (or opt(G) ≤ (1 −

3f(k)k−k+1¹

ε

^′_k

) | V | /n)) on G would enable to decide between opt(I) = n and opt(I) ≤ (1 − ε

^′_k

)n on I.

1.2.2. Positive results from the maximum independent set problem

If we decrease the maximum degree of the graph down to 2, we can easily prove that P

k

P

ARTITION

,

INDUCED

P

k

P

ARTITION

, M

AX

P

k

P

ACKING

and M

IN

k-P

ATH

- P

ARTITION

are polynomial-time computable. The same fact holds for M

AX

W P

k

P

AC

-

KING

(what remains true in forests), although it is a little bit complicated : the proof

consists of a reduction from M

AX

W P

k

P

ACKING

in graphs with maximum degree 2 (resp., in a forest) to the problem of computing a maximum weight independent set in an interval (resp., a chordal) graph, which is known to be polynomial, [FRA 76].

P

ROPOSITION

1.– M

AX

W P

k

P

ACKING

is polynomial in graphs with maximum de- gree 2 and in forests, for any k ≥ 3.

P

ROOF

.– Let I = (G, w) be an instance of M

AX

W P

k

P

ACKING

where G = (V, E) is a graph with maximum degree 2. Hence, G is a collection of disjoint paths or cycles and thus, each connected component may be separately solved. Moreover, wlog., we may assume that each connected component G

^ℓ

of G is a path. Otherwise, a given cycle G

^ℓ

= { v

1

, . . . , v

Nℓ

, v

1

} might be solved by picking the best solution among the solutions computed on the k instances G

^ℓ

\ { [v

1

, v

2

] } , . . . , G

^ℓ

\ { [v

k

, v

k+1

] } . Thus, let G

^ℓ

=

v

₁^ℓ

, . . . , v

^ℓ_Nℓ

be such a path ; we build the instance (H

^ℓ

, w

^ℓ

) of M

AX

WIS where the vertex set of H

^ℓ

corresponds to the paths of length k − 1 in G

^ℓ

: a vertex v is associated to each path P

v

, with weight w

^ℓ

(v) = w(P

v

). Moreover, two vertices u 6 = v are linked in H

^ℓ

iff the corresponding paths P

u

and P

v

share at least one common vertex in the initial graph. We deduce that the set of independent sets in H

^ℓ

corresponds to the set of P

k

in G

^ℓ

. Observe that H

^ℓ

is an interval graph (even a unit interval graph), since each path can be viewed as an interval of the line { 1, · · · , N

^ℓ

} ; hence, H

^ℓ

is chordal. If G is a forest, then any of the graphs H

^ℓ

that correspond to a tree of G is a chordal graph.

1.3. Approximating M

AX

W P

3

P

ACKING

and M

IN

3-P

ATH

P

ARTITION

We present some approximation results for M

AX

W P

3

P

ACKING

and M

IN

3-P

ATH

- P

ARTITION

, that are mainly based on matching and spanning tree heuristics.

1.3.1. M

AX

W P

3

P

ACKING

in graphs of maximum degree 3

For this problem, the best approximate algorithm known so far provides a ra- tio of (

¹₂

− ε), within high (but polynomial) time complexity. This algorithm is de- duced from the one proposed in [ARK 98] to approximate the weighted k-set pa- cking problem for sets of size 3. Furthermore, a simple greedy 1/k-approximation of M

AX

W P

k

P

ACKING

consists in iteratively picking a path of length k − 1 that is of maximum weight. For k = 3 and in graphs of maximum degree 3, the time complexity of this algorithm is between O (n log n) and O (n

²

) (depending on the encoding struc- ture). Actually, in such graphs, one may reach a 1/3-approximate solution, even in time O (α(n, m)n), where α is the inverse Ackerman’s function and m ≤ 3n/2.

T

HEOREM

4.– M

AX

W P

3

P

ACKING

is 1/3 approximable within O (α(n, 3n/2)n) time

complexity in graphs of maximum degree 3 ; this ratio is tight for the algorithm we

analyze.

x x x

y y y z

z z t

T

z

Steps 3.1 and 3.2 T

z

T

t

∪ {[y, t]}

Steps 3.1, 3.2 and 3.3.1

T

y

T

z

Steps 4.1 to 4.3 Figure 1.5. The main configurations of the algorithm

^SubProess

.

P

ROOF

.– The argument uses the following observation : for any spanning tree of maxi- mum degree 3 containing at least 3 vertices, one can build a cover of its edge set into 3 packings of P

3

within linear time. Hence, by computing a maximum-weight spanning tree T = (V, E

T

) on G in O (α(n, 3n/2)n) time, [CHA 00], and by picking the best P

3

-packing among the cover, we obtain a 1/3 approximate solution within an overall time complexity dominated by O (α(n, 3n/2)n).

The construction of the 3 packings P

¹

, P

²

, P

³

is done in the following way : we start with three empty collections P

¹

, P

²

, P

³

and a tree T rooted at r ; according to the degree of r and to the degree of its children, we add some P

3

path P that contains r to the packing P

¹

, remove the edges of P from T , and then recursively repeat this process on the remaining subtrees, alternatively invoking P

²

and P

¹

. This procedure is formally described in the algorithms

^SubProess

(the recursive process) and

^Tree-P₃PakingCover

(the whole process).

Algorithm

^Tree-P3PakingCover

makes an initial call to

^SubProess

, on the whole tree T, rooted on a vertex r that is of degree at most 2 in T . The stopping criterion of the recursive procedure

^SubProess

are the following : the current tree has no edge (then stop), or the current tree is a lonely edge [x, y] ; then add { r

x

, x, y } to P

³

, where r

x

denotes the father of x in T . Concerning the three main configurations of

SubProess

, they are illustrated in Figure 1.5, where T

v

denotes the subtree of T roo- ted at v ; the edges in rigid lines represent the path that is added to the current packing, and the subtrees that are invoked by the recursive calls are indicated.

Tree-P3PakingCover

Input : T = (V

T

, E

T

) spanning tree of maximum degree 3 containing at least 3 vertices and rooted at r such that d

T

(r) ≤ 2.

1 Set P

¹

= P

²

= P

³

= ∅ ;

2 Call

^SubProess

(T

r

, P

¹

, P

²

, P

³

,1) ; 3

^Repair

( P

¹

, P

²

, P

³

) ;

Output ( P

¹

, P

²

, P

³

).

SubProess(

T

x^,

P

^1,

P

^2,

P

^3,

i

⁾

1 If E

Tx

= ∅ then exit ;

Pick y a child of x in T

x

; 2 If E

Tx

= {{ x, y }}

Pick r

x

the father of x in T

r

; 2.1 P

³

←− P

³

∪ {{ r

x

, x, y }} ; exit ;

3 If x is of degree 1 in T

x

Pick z a child of y in T

x

; 3.1 P

ⁱ

←− P

ⁱ

∪ {{ x, y, z }} ;

3.2 Call

^SubProess

(T

z

, P

¹

, P

²

, P

³

,3-i) ; 3.3 If y is of degree 3 in T

x

Pick t the second child of y in T

x

;

3.3.1 Call

^SubProess

( {{ y, t }} ∪ T

t

, P

¹

, P

²

, P

³

, 3-i) ; 4 Else If x is of degree 2 in T

x

Pick z the second child of x in T

x

; 4.1 P

ⁱ

←− P

ⁱ

∪ {{ y, x, z }} ;

4.2 Call

^SubProess

(T

y

, P

¹

, P

²

, P

³

,3-i) ; 4.3 Call

^SubProess

(T

z

, P

¹

, P

²

, P

³

,3-i) ;

At the end of the initial call to

^SubProess

(that is, when the step 2 of

^Tree-P₃^Pa

-

kingCover

has been achieved), P

¹

and P

²

both are packings : one can easily see that

the paths that are added to P

ⁱ

(where i = 1 or i = 2) at a given time t and the ones

that are added again to P

ⁱ

at time t + 2 do not share any common vertex. On the other

hand, P

³

might not be a packing. Let { r

x

, x, y } and { r

x^′

, x

^′

, y

^′

} be two paths from

P

³

such that { r

x

, x, y } ∩ { r

x^′

, x

^′

, y

^′

} 6 = ∅ ; then, either r

x

= r

x^′

, or r

x

= x

^′

. If the

first case occurs, { x, r

x

, x

^′

} has been added to P

ⁱ

(for i = 1 or i = 2), then set : P

ⁱ

=

P

ⁱ

\{{ x, r

x

, x

^′

}} ∪ {{ r

x

, x, y }} and P

³

= P

³

\{{ r

x

, x, y }} . Otherwise, r

x^′

is the fa-

ther of r

x

in T

r

and we have { r

x^′

, r

x

, x } ∈ P

ⁱ

(for i = 1 or i = 2) ; then set : P

ⁱ

=

P

ⁱ

\{{ r

x^′

, r

x

, x }} ∪ {{ r

x^′

, x

^′

, y

^′

}} and P

³

= P

³

\{{ r

x^′

, x

^′

, y

^′

}} . These repairing

operations are made by the algorithm

^Repair

, during step 3 of

^Tree-P₃PakingCover

.

Figure 1.6 provides two examples of the construction of P

¹

, P

²

and P

³

. The overall

time complexity of

^Tree-P3PakingCover

is linear : first, the number of recursive

calls to

^SubProess

may not exceed 2/3n and second, |P

³

| is at most O (log n).

iteration 1

∅ ∅ ∅

P2

P1 P3 repair

P1

P2

P3

P1

P2

P3 repair

P1

P2

P3

∅

∅ iteration 2

iteration 1 iteration 2 iteration 3

∅

∅ T2

T1

remaining subtrees iteration 3 remaining subtrees

remaining subtrees

Figure 1.6. Two examples of the construction of the 3 packings P

ⁱ

for i = 1, 2, 3.

Repair(

P

^1,

P

^2,

P

³⁾

1 For any (P = { r

x

, x, y } 6 = P

^′

= { r

x^′

, x

^′

, y

^′

} ) ∈ P

³

s.t. r

x

= r

x^′

Set i ∈ { 1, 2 } s.t. { x, r

x

, x

^′

} ∈ P

ⁱ

;

1.1 P

ⁱ

←− P

ⁱ

\{{ x, r

x

, x

^′

}} ∪ {{ r

x

, x, y }} ; P

³

←− P

³

\{{ r

x

, x, y }} ; 2 For any (P = { r

x

, x, y } 6 = P

^′

= { r

x^′

, x

^′

, y

^′

} ) ∈ P

³

s.t. r

x

= x

^′

Set i ∈ { 1, 2 } s.t. { r

x^′

, r

x

, x } ∈ P

ⁱ

;

2.1 P

ⁱ

←− P

ⁱ

\{{ r

x^′

, r

x

, x }} ∪ {{ r

x^′

, x

^′

, y

^′

}} ; P

³

←− P

³

\{{ r

x^′

, x

^′

, y

^′

}} ; Output ( P

¹

, P

²

, P

³

).

We now can deduce an approximate algorithm

^MaxW

P

3^Paking

, that consists in com-

puting a P

3

-packing cover ( P

¹

, P

²

, P

³

) of a maximum spanning tree of G, and

then picking the best collection among ( P

¹

, P

²

, P

³

). This algorithm provides a 1/3-

approximation within O (α(n, 3n/2)n) time complexity (the overall complexity of the

algorithm is dominated by the one of computing the initial spanning tree). Concerning the approximation level, consider that the weight w(T ) of a maximum spanning tree T is at least the weight of an optimal P

3

-packing, since any P

3

packing can be com- pleted into a spanning tree (if the input graph is connected). Then the result is trivial (let P

^∗

denote an optimal solution) :

w( P ) ≥ 1/3 w( P

¹

) + w( P

²

) + w( P

³

)

≥ 1/3w(T ) ≥ 1/3w( P

^∗

) The proof of tightness is omitted.

1.3.2. M

AX

W P

3

P

ACKING

in bipartite graphs of maximum degree 3

If we restrict ourselves to bipartite graphs, we slightly improve the ratio of

¹₂

− ε, [ARK 98] up to

¹₂

. We then show that, in the unweighted case, this result holds wi- thout any constraint on the graph maximum degree. The key idea here is to trans- form the problem of finding a P

3

Packing in the initial bipartite graph G = (L, R; E) into the problem of finding a maximum matching in two graphs G

L

and G

R

, where G

L

(resp., G

R

) contains the representative edge of the P

3

of the initial graph with their two extremities in L (resp., in R). Formally, from an instance I = (G, w) of M

AX

W P

3

P

ACKING

, where G = (L, R; E) is a bipartite graph of maximum degree 3, we build two weighted graphs (G

L

, w

L

) and (G

R

, w

R

), where G

L

= (L, E

L

) and G

R

= (R, E

R

). Two vertices x 6 = y from L are linked in G

L

iff there exists in G a path P

x,y

of length 2 from x to y : [x, y] ∈ E

L

iff ∃ z ∈ R s.t. [x, z], [z, y] ∈ E. The weight w

L

(x, y) is defined as w

L

(x, y) = max { w(x, z) + w(z, y) | [x, z], [z, y] ∈ E } . The weighted graph (G

R

, w

R

) is defined by considering R instead of L. If G is of maximum degree 3, then the following fact holds :

P

ROPERTY

2.– From any matching M on G

L

(resp., on G

R

), one can deduce a P

3

packing P

^M

of weight w( P

^M

) = w

L

(M ) (resp., w( P

^M

) = w

R

(M )), where G is of degree at most 3.

P

ROOF

.– Let M be a matching on G

L

, and P

^M

the corresponding P

3

collection on G. Suppose that two paths P

x,y

6 = P

x^′,y^′

∈ P

^M

share a common vertex t. Because M is a matching, we have { x, y } ∩ { x

^′

, y

^′

} = ∅ ; hence, the vertex t belongs to R and is the internal vertex of both P

x^′,y^′

and P

x^′,y^′

, which contradicts the assumption on the graph maximum degree.

In light of this fact, we propose the algorithm

^{Weighted P}₃^-Paking

that consists

in computing two maximum matchings on G

L

and G

R

, and then picking the best cor-

responding packing in G. The time complexity of this algorithm is mainly the time

complexity of computing a maximum weight matching in graphs of maximum degree

9, that is O ( | V |

²

log | V | ), [LOV 86].

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

n n n n n n n n

1

Figure 1.7. Tightness of

^{Weighted P}3^-Paking

analysis.

Weighted P3^-Paking

1 Build the weighted graphs (G

L

, w

L

) and (G

R

, w

R

) ;

2 Compute a maximum weight matching M

_L^∗

(resp., M

_R^∗

) on (G

L

, w

L

) (resp., on (G

R

, w

R

)) ;

3 Deduce from M

_L^∗

(resp., from M

_R^∗

) a P

3

packing P

^L

(resp., P

^R

) according to Property 2 ;

4 Output the best packing P among P

^L

and P

^R

.

T

HEOREM

5.–

^{Weighted P}₃^-Paking

provides a 1/2-approximation for M

AX

W P

3

- P

ACKING

in bipartite graphs with maximum degree 3 and this ratio is tight.

P

ROOF

.– Let P

^∗

be an optimum P

3

-packing on I = (G, w), we denote by P

L^∗

(resp., by P

R^∗

) the paths of P

^∗

of which the two endpoints belong to L (resp., to R) ; thus, opt(I) = w( P

L^∗

) + w( P

L^∗

). For any path P = P

x,y

∈ P

L^∗

, [x, y] is an edge from E

L

, of weight w

L

(x, y) ≥ w(P

x,y

). Hence, M

L

= { [x, y] | P

x,y

∈ P

L^∗

} is a matching on G

L

that satisfies :

w

L

(M

L

) ≥ w( P

L^∗

) [1.1]

Moreover, since M

_L^∗

is a maximum weight matching on G

L

, we have w

L

(M

L

) ≤ w

L

(M

_L^∗

). Thus, using inequality [1.1] and Property 2 (and by applying the same ar- guments on G

R

), we deduce :

w( P

^L

) ≥ w( P

L^∗

), w( P

^R

) ≥ w( P

R^∗

) [1.2]

Finally, the solution output by the algorithm satisfies w( P ) ≥ 1/2(w( P

^L

) + w( P

^R

))

and we directly deduce from inequalities [1.2] the expected result. The instance I =

(G, w) that provides the tightness is depicted in Figure 1.7. It consists of a graph on

12n vertices on which one can easily observe that w( P

^L

) = w( P

^R

) = 2n(n + 2) and

w( P

^∗

) = 2n(2n + 2).

Concerning the unweighted case, we may obtain the same performance ratio wi- thout the restriction on the graph maximum degree. The main differences compared to the previous algorithm lie in the construction of the two graphs G

L

, G

R

: starting from G, we duplicate each vertex r

i

∈ R by adding a new vertex r

_i^′

with the same neighborhood as r

i

(this operation, often called multiplication of vertices in the litera- ture, is used in the characterization of perfect graphs). We then add the edge [r

i

, r

_i^′

]. If R

L

denotes the vertex set { r

i

, r

_i^′

| r

i

∈ R } , the following properties hold :

P

ROPERTY

3.–

(i) From any matching M on G

L

, one can deduce a matching M

^′

of cardinality

| M

^′

| ≥ | M | on G

L

that saturates R

L

.

(ii) From any matching M on G

L

(resp., on G

R

) that saturates R

L

(resp., L

R

), one can deduce a P

3

packing P

^M

on G of size |P

^M

| = | M | − | R | .

P

ROOF

.– For (i). Let M be a matching on G

L

and consider a given vertex r

i

∈ R.

If M contains no edge incident to { r

i

, r

^′_i

} , then add [r

i

, r

_i^′

] to M ; if M contains an edge e incident to r

i

(resp., to r

_i^′

), but no edge incident to r

^′_i

(resp., to r

i

), then set M = M \{ e } ∪ { [r

i

, r

_i^′

] } .

For (ii). Let M be a matching on G

L

that saturates R

L

, we respectively denote by J the set of vertices r

i

∈ R such that [r

i

, r

_i^′

] ∈ M and by p = | J | its cardinality.

We consider the matching M

^′

deduced from M by deleting the edges [r

i

, r

_i^′

] ; hence,

| M

^′

| = | M | − p. From the fact that M saturates R

L

, we first deduce that | M | =

| R

L

| − p = 2 | R | − p ; we then observe that, for any vertex r

i

∈ / J , there exists two edges [l

_i¹

, r

i

] and [l

_i²

, r

^′

i] in M

^′

, that define the P

³

P

i

= { l

_i¹

, r

i

, l

²_i

} of the initial graph G. The collection P

^M

= ∪

ri∈J/

{ P

i

} obviously is a P

³

packing of size | M

^′

| /2 on G.

One just has to obverse that | M

^′

| = 2 | R | − 2p = 2( | M | − | R | ) in order to conclude.

P3^-Paking

1 Build the graph G

L

(resp., G

R

) obtained from G by multiplication of vertices on R (resp., on L) ;

2 Compute a maximum size matching M

L

(resp., M

R

) on G

L

(resp., on G

R

) ; According to Property 3 item (i), deduce from M

L

(resp., from M

R

) a maximum size matching M

_L^∗

(resp., M

_R^∗

) that saturates R

L

(resp., L

R

) ;

3 According to Property 3 item (ii), deduce from M

_L^∗

(resp., from M

_R^∗

) a P

3

packing P

^L

(resp., P

^R

) of size | M

_L^∗

| − | R | (resp., | M

_R^∗

| − | L | ) ; 4 Output the best packing P among P

^L

and P

^R

.

The approximate algorithm

^P₃^-Paking

works as previously, except that we com-

pute a maximum (size) matching M

_L^∗

(resp., M

_R^∗

) on G

L

(resp., G

R

) that saturates R

L

(resp., L

R

) in step 2, and that the P

3

packing P

^L

(resp., P

^R

) is obtained from M

_L^∗

(resp., M

_R^∗

) by deleting the edges [r

i

, r

_i^′

] (resp., [l

i

, l

_i^′

]) in step 3.

T

HEOREM

6.–

^P3^-Paking

provides a 1/2-approximation for M

AX

P

3

P

ACKING

in bipartite graphs and this ratio is tight. The time complexity of this algorithm is O (m √ n).

P

ROOF

.– Let P

L^∗

= { P

1

, · · · , P

q

} be the set of paths from the optimal solution having their two endpoints in L ; P

L^∗

can easily be converted on G

L

into a matching M of size | M | = 2q + ( | R | − q) = |P

L^∗

| + | R | . From the optimality of M

_L^∗

on G

L

, we deduce that | M

_L^∗

| ≥ | M | and hence, that |P

^L

| ≥ |P

L^∗

| . The same obviously holds for P

R^∗

and the result is immediate. The time complexity of the unweighted version of the algorithm still is dominated by the one of computing a maximum (size) matching, that is O (m √

n), [LOV 86]. The proof of tightness is omitted.

1.3.3. M

IN

3-P

ATH

P

ARTITION

in general graphs

To our knowledge, the approximability of M

IN

k-P

ATH

P

ARTITION

(or M

IN

P

A

-

TH

P

ARTITION

) has not been studied so far. Here, we propose a 3/2-approximation for M

IN

3-P

ATH

P

ARTITION

. Although this problem can be viewed as an instance of 3-set cover (view the set of all paths of length 0, 1 or 2 in G as sets on V ), M

IN

3- P

ATH

P

ARTITION

and the minimum 3-set cover problem are different. For instance, consider a star K

1,2n

; the optimum value of the corresponding 3-set cover instance is n, whereas the optimum value of the 3-path partition is 2n − 1. Note that, concer- ning M

IN

P

ATH

P

ARTITION

(that is, the approximation of ρ(G)), we can trivially see that it is not (2 − ε)-approximable, from the fact that deciding whether ρ(G) = 1 or ρ(G) ≥ 2 is NP-complete. Actually, we can more generally establish that ρ(G) is not in APX : otherwise, we could obtain a PTAS for the traveling salesman pro- blem with weight 1 and 2 when opt(I) = n, which is not possible, unless P=NP.

The algorithm

^{Minimum 3Path Partition}

we propose runs in two phases : first, it computes a maximum matching M

₁^∗

on the input graph G = (V, E) ; then, it matches through M

₂^∗

a maximum number of edges from M

₁^∗

to vertices from V \ M

₁^∗

. Those two matchings define the P

3

and the P

2

of the approximate solution.

T

HEOREM

7.–

^{Minimum 3Path Partition}

provides a 3/2-approximation for M

IN

3- P

ATH

P

ARTITION

in general graphs within O (nm + n

²

log n) time and this ratio is tight.

P

ROOF

.– Let G = (V, E) be an instance of M

IN

3-P

ATH

P

ARTITION

. Let P

^∗

= ( P

2^∗

, P

1^∗

, P

0^∗

) and P

^′

= ( P

2^′

, P

1^′

, P

0^′

) respectively be an optimal solution and the ap- proximate 3-path partition on G, where P

i^∗

and P

i^′

denote for i = 0, 1, 2 the set of paths of length i. By construction of the approximate solution, we have :

apx(I) = | V | − | M

₁^∗

| − | M

₂^∗

| [1.3]